Need to Fix Broken Edge Loop?
An alternative, Sub-D approach: absolutely no better than yours, but with all the tools in Blender that are geared to working this way, personally I find this kind of pattern  gives rise  to fewer mysterious faults in the mesh, and is easier to tweak later.



Starting with a bevelled pair of drawers:


CtrlR  Cut in 3 edge-loops in both directions
F Fill the central faces
AltP Poke the filled faces
CtrlShiftB bevel the central vertices
Select both resulting faces, and Loop-Tools > Circle, ( or Space-Bar > Circle) setting the Radius field as desired.
Inset I and Extrude E the circular faces, to create holding loops and the recess.
Assign a Subdivision Surface modifier.